Blowup driven by critical balance in a differential kinetic model of gravity wave turbulence

Abstract

We describe the blowup scenarios in a phase-parametrized differential approximation kinetic model (N-DAM), inspired by the physics of deep water surface gravity waves and recently obtained using large-$N$ summation techniques under a local approximation in wavenumber space. Previous work showed that the model admits steady-state solutions interpolating between the Kolmogorov-Zakharov spectrum $E(\omega )\propto {\omega }^{-4}$ and either a strong-turbulence regime $E(\omega )\propto {\omega }^{-2}$ or the Phillips critical-balance spectrum $E(\omega )\propto {\omega }^{-5}$ at small scales. These solutions reproduce scaling regimes expected in gravity-wave kinetics, suggesting that the N-DAM may serve as an effective augmented version of an earlier differential approximation model introduced by Hasselmann. Here we investigate finite-time blowup in the N-DAM and show that it is generically governed by the critical-balance regime. This leads to a non-Kolmogorov finite-time transfer of the energy from the IR towards the UV for any value of the parameter $\varphi \in [0,\pi )$. We observe a bifurcation in the blowup dynamics from continuous to discrete self-similarity as $\varphi$ is increased above a critical value ${\varphi }_{\ast }\simeq 2.7$. To our knowledge, this is the first example of a discretely self-similar blowup in the kinetic theory of waves.